Think of a disk image like a box, which contains files. Mounting it means essentially opening the box. Double clicking on a disk image mounts it, and it will then appear in the sidebar of finder. From there you can copy the files wherever you need. After which you can unmount (eject) the disk image and delete it.

It means a drive or partition (flash drive, hard drive, ect) is active. When you plug in your flash drive into your computer and it appears on the desktop it is mounted. You can actively use it. When you eject it, even though it is still plugged in, it is unmounted (no longer active).

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